Thomas Dube.

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Font size Robotics Research
l^hnical Report

On the Existence of Generic Coordinates

by
Thomas Dubd

Technical Report No. 456

Robotics Report No. 203

June, 1989

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Hterm((/.,(/i)) > i^+[P .
D

Notation: For / an ideal of A' and P G PP,, let Np{I,d) denote the
number of distinct head terms of the form x^~'^x^^iP which belong to poly-
nomials in /. That is,

Though it will not be used in this report, one might note that that is the
Hilbert function ,{fi), . . ‚Ė† ,(f>i{fm) must also be linearly indepen-
dent. Triangulating the 4>,{fj), it is possible to create linear combinations
gi,...,gm '^ J oi the form

m

m

; = i

such that a, J ‚ā¨ K{y), and the head terms of the 9, are distinct. Then
Qi ‚ÄĒ ,{h,) where h, = J2T=i ^'.ifj- Each h, has a head term of the form
xf~'^x^^.iP, and thus by the preceding lemma, Hterm(g,) is therefore also
of this form. And so each of the gi, ‚Ė† . ‚Ė† ,gm ^ J has a distinct head term of
the form xf~'^x^^iP and therefore Np{J,d) > c.

A symmetric argument employing the inverse homomorphism (f>J^ shows
that Np{I',d) > Np{J,d), completing the proof of the lemma.
D

Lemma 4 Let I ¬£ A, and I' be the ideal generated by I in A'. Suppose
that I' contains a polynomial h with Hterm((;/i, (/i)) = i^'^x^^jP. Then
Np{I,d) >c + l.

Proof. Since Hterm((^,(/i)) = i^-'x^+iP,

d

,{h) = J^a,x'l-'x',^,P + L.O.T.

¬ę=c

Using the inverse transformation (pj^, h can be written as:

d

h = (^a.x^-(x,,i-yxJ')P + L.O.T.

- i:¬ę.(E(-ir'( ‚ÄĘ )y'-x^v,^,)p+ L.O.T.

3 THE CHANGE OF COORDINATES 7

The a, are elements of K{y) and hence can be written as a, = Y^T=m^i,2y'-
Collecting ternis around powers of y

^ = Ej/^(EI:^-.+;(-1)-M ‚ÄĘ ]x''r^xU,)P + h.O.T.

z=m ,-c ;=0 \ -' /

00 d d ( \

The polynomial h has been written in the form k = J^V^^z- From the
properties of polynomial rings with extended coefficient fields, it follows
that each k^ G /. Let r = min{2|3, such that 6,_2_, / 0}. Then, for
q = r - d,...,r,

K = i:x^V,^,(y:(-in( ;. )6.,,_.>,)P + L.0.T.

If y < r ‚ÄĒ 9 then q ‚ÄĒ i + j;)P + L.0.T.

The largest power product remaining in the expression for /i, is x^'^'^'^x^^^P,
and it's coefficient is

Hcoef(/t,) = E(-ir^^' (,!,)''¬ę,-

1 = C

= E(-i)-'-L" A.

where /?, denotes the value 6, ,.- 1- Consider the /?, to be variables. There
are (f ‚ÄĒ c + 1 of these variables:

3 THE CHANGE OF COORDINATES 8

Now the d+ 1 expressions Hcoef(/ir_d), . ‚ÄĘ ‚ÄĘ ,Hcoef(/ir) are linearly indepen-
dent over these variables. Therefore, fixing the values of any d ‚ÄĒ c + I of
the Hcoef(/ifc) has the effect of fixing the values of the /9,'s. Therefore, if
d ‚ÄĒ c + 1 of these expressions evaluate to zero, then all of the /?, must be
zero. This contradicts the choice of r as the least value v^'here at least one
of the 0, is non-zero. Therefore, at least c+ 1 of the Hcoef(/ii) are non-zero,
and Head(7) contains at least c + I different power products of the form

D

Corollary 5 If Np{4>,{r),d) = c, then x'^^-^4+iP ^ Head(

1

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